Showing SO(3) Subset is Projective Plane Diffeomorphic

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I was wondering if anyone can help me to show that the subset of SO(3) contaning all
matrices A with det(A+id)=0 is a submanifold diffeomorphic to real projective plane.
Thanks.
 
Well, what ideas have you had on it so far? Where are you stuck?
 
Actually, I could show that SO(3) is homeo. to RP^3. But on don't know why the condition det(A+id)=0 implies that this subset would be diffeo. to RP^2.
 

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